Players Have a "Rule 0" Too
To the best of my knowledge, there is nothing specifically in the rules for making a player mute, and certainly not anything that trades speech for a bonus to skills, let alone Sleight of Hand. However, that shouldn't limit you from being able to play this concept. An oft-overlooked section in the Player's Handbook is the "Customizing Your Character" segment. It explicitly states that you can call any of your feats, features, or skills whatever you want. As long as the mechanics are the same, it doesn't matter what you call it.
You can call your skills, feats, and class features whatever your character would call them. Lidda, the halfling rogue, talks about “footpaddin’” rather than about “moving silently,” so her player writes “Footpaddin’” down on her character sheet to stand for the Move Silently skill. Ember, the monk, calls her Move Silently skill “Rice Paper Walk.”
Enter the Pathetic flaw, from Unearthed Arcana. All the description says is that you are weaker in a given attribute (lit; "You are weaker in an attribute than you should be."). Exactly why you're weaker is left up to the player to define. So, you can use this flaw to simulate a wide range of disabilities. A penalty to Charisma would certainly work to show the difficulty someone has in using communication-based skills without being able to vocalize (in other words, using only sign and body language). Leaving it alone fits what little fluff it is, but as long as the mechanics are correct, the rules not only don't prevent, but in fact encourage having it on your character sheet as "Mute".
In exchange, you could take Skill Focus (Sleight of Hand) if you want to specialize. Or, for a more general approach, something like Deft Hands would work.
"But that doesn't actually make me mute!"
As I said to begin with, muteness isn't something that the rules cover properly. However, your character is what you say they are. If you declare your character mute, never speak in character, and instead communicate through sign language, then your character is mute. Recording the flaw you take as being muteness can help serve to remind you, and to make things clear to other players, as it gives you something on paper you can point to, but it isn't even necessary by the rules. All that's really necessary is for you to state them so. The flaw for feat trade simply serves to get you the bonus you want. Otherwise, you could have a character that is mute, but has developed effective ways of coping simply by not taking any mechanical penalty. The important thing is simply to roleplay it, because at the end of the day D&D is a roleplaying game, not just a dice-rolling game.
I'm sorry if this isn't the answer you're looking for, but using only material from published sources, it is likely to be the closest you can come, and is certainly the one likely to be legal in the most campaigns.